Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces

Abstract

Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Suppose that 𝑇∶𝐾→2^𝐾 is a multivalued strictly pseudocontractive mapping such that 𝐹(𝑇)≠∅. A Krasnoselskii-type iteration sequence {𝑥_𝑛} is constructed and shown to be an approximate fixed point sequence of 𝑇; that is, lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0 holds. Convergence theorems are also proved under appropriate additional conditions.

1. Introduction

For several years, the study of fixed point theory of multivalued nonlinear mappings has attracted, and continues to attract, the interest of several well-known mathematicians (see, e.g., Brouwer [1], Kakutani [2], Nash [3, 4], Geanakoplos [5], Nadler [6], and Downing and Kirk [7]).

Interest in such studies stems, perhaps, mainly from the usefulness of such fixed point theory in real-world applications, such as in Game Theory and Market Economy, and in other areas of mathematics, such as in Nonsmooth Differential Equations. We describe briefly the connection of fixed point theory of multivalued mappings and these applications.

Game Theory and Market Economy. In game theory and market economy, the existence of equilibrium was uniformly obtained by the application of a fixed point theorem. In fact, Nash [3, 4] showed the existence of equilibria for noncooperative static games as a direct consequence of Brouwer [1] or Kakutani [2] fixed point theorem. More precisely, under some regularity conditions, given a game, there always exists a multivalued mapping whose fixed points coincide with the equilibrium points of the game. A model example of such an application is the Nash equilibrium theorem (see, e.g., [3]).

Consider a game 𝐺=(𝑢_𝑛,𝐾_𝑛) with 𝑁 players denoted by 𝑛, 𝑛=1,…,𝑁, where 𝐾_𝑛⊂ℝ^𝑚𝑛 is the set of possible strategies of the 𝑛th player and is assumed to be nonempty, compact, and convex, and 𝑢_𝑛∶𝐾∶=𝐾₁×𝐾₂⋯×𝐾_𝑁→ℝ is the payoff (or gain function) of the player 𝑛 and is assumed to be continuous. The player 𝑛 can take individual actions, represented by a vector 𝜎_𝑛∈𝐾_𝑛. All players together can take a collective action, which is a combined vector 𝜎=(𝜎₁,𝜎₂,…,𝜎_𝑁). For each 𝑛, 𝜎∈𝐾 and 𝑧_𝑛∈𝐾_𝑛, we will use the following standard notations: 𝐾_−𝑛∶=𝐾₁×⋯×𝐾_𝑛−1×𝐾_𝑛+1×⋯×𝐾_𝑁,𝜎_−𝑛∶=𝜎₁,…,𝜎_𝑛−1,𝜎_𝑛+1,…,𝜎_𝑁,𝑧_𝑛,𝜎_−𝑛∶=𝜎₁,…,𝜎_𝑛−1,𝑧_𝑛,𝜎_𝑛+1,…,𝜎_𝑁. (1) A strategy 𝜎_𝑛∈𝐾_𝑛 permits the 𝑛’th player to maximize his gain under the condition that the remaining players have chosen their strategies 𝜎_−𝑛 if and only if 𝑢_𝑛𝜎_𝑛,𝜎_−𝑛=max_{𝑧𝑛∈𝐾𝑛}𝑢_𝑛𝑧_𝑛,𝜎_−𝑛. (2) Now, let 𝑇_𝑛∶𝐾_−𝑛→2^𝐾𝑛 be the multivalued mapping defined by 𝑇_𝑛𝜎_−𝑛∶=Argmax_{𝑧𝑛∈𝐾𝑛}𝑢_𝑛𝑧_𝑛,𝜎_−𝑛∀𝜎_−𝑛∈𝐾_−𝑛. (3)

Definition 1. A collective action 𝜎=(𝜎₁,…,𝜎_𝑁)∈𝐾 is called a Nash equilibrium point if, for each 𝑛, 𝜎_𝑛 is the best response for the 𝑛’th player to the action 𝜎_−𝑛 made by the remaining players. That is, for each 𝑛, 𝑢_𝑛(𝜎)=max_{𝑧𝑛∈𝐾𝑛}𝑢_𝑛𝑧_𝑛,𝜎_−𝑛 (4) or, equivalently, 𝜎_𝑛∈𝑇_𝑛𝜎_−𝑛. (5) This is equivalent to that 𝜎 is a fixed point of the multivalued mapping 𝑇∶𝐾→2^𝐾 defined by 𝑇(𝜎)∶=𝑇₁𝜎₋₁×𝑇₂𝜎₋₂×⋯×𝑇_𝑁𝜎_−𝑁. (6) From the point of view of social recognition, game theory is perhaps the most successful area of application of fixed point theory of multivalued mappings. However, it has been remarked that the applications of this theory to equilibrium are mostly static: they enhance understanding conditions under which equilibrium may be achieved but do not indicate how to construct a process starting from a nonequilibrium point and convergent to equilibrium solution. This is part of the problem that is being addressed by iterative methods for fixed point of multivalued mappings.

Nonsmooth Differential Equations. The mainstream of applications of fixed point theory for multivalued mappings has been initially motivated by the problem of differential equations (DEs) with discontinuous right-hand sides which gave birth to the existence theory of differential inclusion (DIs). Here is a simple model for this type of application.

Consider the initial value problem 𝑑𝑢()[](0)𝑑𝑡=𝑓𝑡,𝑢,a.e.𝑡∈𝐼∶=−𝑎,𝑎,𝑢=𝑢₀. (7) If 𝑓∶𝐼×ℝ→ℝ is discontinuous with bounded jumps, measurable in 𝑡, one looks for solutions in the sense of Filippov [8, 9] which are solutions of the differential inclusion 𝑑𝑢()(0)𝑑𝑡∈𝐹𝑡,𝑢,a.e.𝑡∈𝐼,𝑢=𝑢₀, (8) where 𝐹(𝑡,𝑥)=liminf_𝑦→𝑥𝑓𝑡,𝑦,limsup_𝑦→𝑥𝑓𝑡,𝑦. (9) Now set 𝐻∶=𝐿²(𝐼) and let 𝑁_𝐹∶𝐻→2^𝐻 be the multivalued NemyTskii operator defined by 𝑁_𝐹(𝑢)∶={𝑣∈𝐻∶𝑣(𝑡)∈𝐹(𝑡,𝑢(𝑡))a.e.𝑡∈𝐼}. (10) Finally, let 𝑇∶𝐻→2^𝐻 be the multivalued mapping defined by 𝑇∶=𝑁_𝐹𝑜𝐿⁻¹, where 𝐿⁻¹ is the inverse of the derivative operator 𝐿𝑢=𝑢^′ given by 𝐿⁻¹𝑣(𝑡)∶=𝑢₀+^𝑡₀𝑣(𝑠)𝑑𝑠. (11) One can see that problem (8) reduces to the fixed point problem: 𝑢∈𝑇𝑢.

Finally, a variety of fixed point theorems for multivalued mappings with nonempty and convex values is available to conclude the existence of solution. We used a first-order differential equation as a model for simplicity of presentation, but this approach is most commonly used with respect to second-order boundary value problems for ordinary differential equations or partial differential equations. For more about these topics, one can consult [10–13] and references therein as examples.

We have seen that a Nash equilibrium point is a fixed point 𝜎 of a multivalued mapping 𝑇∶𝐾→2^𝐾, that is, a solution of the inclusion 𝑥∈𝑇𝑥 for some nonlinear mapping 𝑇. This inclusion can be rewritten as 0∈𝐴𝑥, where 𝐴∶=𝐼−𝑇 and 𝐼 is the identity mapping on 𝐾.

Many problems in applications can be modeled in the form 0∈𝐴𝑥, where, for example, 𝐴∶𝐻→2^𝐻 is a monotone operator,thatis,⟨𝑢−𝑣,𝑥−𝑦⟩≥0 for all 𝑢∈𝐴𝑥, 𝑣∈𝐴𝑦, 𝑥,𝑦∈𝐻. Typical examples include the equilibrium state of evolution equations and critical points of some functionals defined on Hilbert spaces 𝐻. Let 𝑓∶𝐻→(−∞,+∞] be a proper, lower-semicontinuous convex function; then it is known (see, e.g., Rockafellar [14] or Minty [15]) that the multivalued mapping 𝑇∶=𝜕𝑓, the subdifferential of 𝑓, is maximal monotone, where for 𝑤∈𝐻, (𝑥)𝑦(𝑥)≥⟨⟩.𝑤∈𝜕𝑓⟺𝑓−𝑓𝑦−𝑥,𝑤∀𝑦∈𝐻⟺𝑥∈Argmin𝑓−⋅,𝑤 (12) In this case, the solutions of the inclusion 0∈𝜕𝑓(𝑥), if any, correspond to the critical points of 𝑓, which are exactly its minimizer points.

Also, the proximal point algorithm of Martinet [16] and Rockafellar [17] studied also by a host of authors is connected with iterative algorithm for approximating a solution of 0∈𝐴𝑥 where 𝐴 is a maximal monotone operator on a Hilbert space.

In studying the equation 𝐴𝑢=0, Browder introduced an operator 𝑇 defined by 𝑇∶𝐼−𝐴 where 𝐼 is the identity mapping on 𝐻. He called such an operator 𝑇pseudocontractive. It is clear that solutions of 𝐴𝑢=0 now correspond to fixed points of 𝑇. In general, pseudocontraactive mappings are not continuous. However, in studying fixed point theory for pseudocontractive mappings, some continuity condition (e.g., Lipschitz condition) is imposed on the operator. An important subclass of the class of Lipschitz pseudocontractive mappings is the class of nonexpansive mappings, that is, mappings 𝑇∶𝐾→𝐾 such that ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾. Apart from being an obvious generalization of the contraction mappings, nonexpansive mappings are important, as has been observed by Bruck [18], mainly for the following two reasons. (i)Nonexpansive mappings are intimately connected with the monotonicity methods developed since the early 1960s and constitute one of the first classes of nonlinear mappings for which fixed point theorems were obtained by using the fine geometric properties of the underlying Banach spaces instead of compactness properties. (ii)Nonexpansive mappings appear in applications as the transition operators for initial value problems of differential inclusions of the form 0∈(𝑑𝑢/𝑑𝑡)+𝑇(𝑡)𝑢, where the operators {𝑇(𝑡)} are, in general, set-valued and are accretive or dissipative and minimally continuous. The class of strictly pseudocontractive mappings defined in Hilbert spaces which was introduced in 1967 by Browder and Petryshyn [19] is a superclass of the class of nonexpansive mappings and a subclass of the class of Lipschitz pseudocontractions. While pseudocontractive mappings are generally not continuous, the strictly pseudocontractive mappings inherit Lipschitz property from their definitions. The study of fixed point theory for strictly pseudocontractive mappings may help in the study of fixed point theory for nonexpansive mappings and for Lipschitz pseudocontractive mappings. Consequently, the study by several authors of iterative methods for fixed point of multivalued nonexpansive mappings has motivated the study in this paper of iterative methods for approximating fixed points of the more general strictly pseudocontractive mappings. Part of the novelty of this paper is that, even in the special case of multivalued nonexpansive mappings, convergence theorems are proved here for the Krasnoselskii-type sequence which is known to be superior to the Mann-type and Ishikawa-type sequences so far studied. It is worth mentioning here that iterative methods for approximating fixed points of nonexpansive mappings constitute the central tools used in signal processing and image restoration (see, e.g., Byrne [20]).

Let 𝐾 be a nonempty subset of a normed space 𝐸. The set 𝐾 is called proximinal (see, e.g., [21–23]) if for each 𝑥∈𝐸 there exists 𝑢∈𝐾 such that 𝑑(𝑥,𝑢)=inf𝑥−𝑦∶𝑦∈𝐾=𝑑(𝑥,𝐾), (13) where 𝑑(𝑥,𝑦)=‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐸. Every nonempty, closed, and convex subset of a real Hilbert space is proximinal. Let 𝐶𝐵(𝐾) and 𝑃(𝐾) denote the families of nonempty, closed, and bounded subsets and of nonempty, proximinal, and bounded subsets of 𝐾, respectively. The Hausdorff metric on 𝐶𝐵(𝐾) is defined by 𝐷(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝑏,𝐴) (14) for all 𝐴,𝐵∈𝐶𝐵(𝐾). Let 𝑇∶𝐷(𝑇)⊆𝐸→𝐶𝐵(𝐸) be a multivalued mapping on 𝐸. A point 𝑥∈𝐷(𝑇) is called a fixed point of 𝑇 if 𝑥∈𝑇𝑥. The fixed point set of 𝑇 is denoted by 𝐹(𝑇)∶={𝑥∈𝐷(𝑇)∶𝑥∈𝑇𝑥}.

A multivalued mapping 𝑇∶𝐷(𝑇)⊆𝐸→𝐶𝐵(𝐸) is called 𝐿-Lipschitzian if there exists 𝐿>0 such that 𝐷𝑇𝑥,𝑇𝑦≤𝐿𝑥−𝑦∀𝑥,𝑦∈𝐷(𝑇). (15) When 𝐿∈(0,1) in (15), we say that 𝑇 is a contraction, and 𝑇 is called nonexpansive if 𝐿=1.

Several papers deal with the problem of approximating fixed points of multivalued nonexpansive mappings (see, e.g., [21–26] and the references therein) and their generalizations (see, e.g., [27, 28]).

Sastry and Babu [21] introduced the following iterative schemes. Let 𝑇∶𝐸→𝑃(𝐸) be a multivalued mapping, and let 𝑥^∗ be a fixed point of 𝑇. Define iteratively the sequence {𝑥_𝑛}_𝑛∈ℕ from 𝑥₀∈𝐸 by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑦_𝑛,𝑦_𝑛∈𝑇𝑥_𝑛,𝑦_𝑛−𝑥^∗=𝑑𝑇𝑥_𝑛,𝑥^∗, (16) where 𝛼_𝑛 is a real sequence in (0,1) satisfying the following conditions:(i)∑^∞_𝑛=1𝛼_𝑛=∞,(ii)lim𝛼_𝑛=0.

They also introduced the following scheme: 𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑧_𝑛,𝑧_𝑛∈𝑇𝑥_𝑛,𝑧_𝑛−𝑥^∗=𝑑𝑥^∗,𝑇𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑢_𝑛,𝑢_𝑛∈𝑇𝑦_𝑛,𝑢_𝑛−𝑥^∗=𝑑𝑇𝑦_𝑛,𝑥^∗, (17) where {𝛼_𝑛} and {𝛽_𝑛} are sequences of real numbers satisfying the following conditions:(i)0≤𝛼_𝑛,𝛽_𝑛<1,(ii)lim_𝑛→∞𝛽_𝑛=0,(iii) ∑^∞_𝑛=1𝛼_𝑛𝛽_𝑛=∞. Sastry and Babu called a process defined by (16) a Mann iteration process and a process defined by (17) where the iteration parameters 𝛼_𝑛 and 𝛽_𝑛 satisfy conditions (i), (ii), and (iii) an Ishikawa iteration process. They proved in [21] that the Mann and Ishikawa iteration schemes for a multivalued mapping 𝑇 with fixed point 𝑝 converge to a fixed point of 𝑇 under certain conditions. More precisely, they proved the following result for a multivalued nonexpansive mapping with compact domain.

Theorem SB (Sastry and Babu [21]). Let 𝐻 be real Hilbert space, let 𝐾 be a nonempty, compact, and convex subset of 𝐻, and let 𝑇∶𝐾→𝐶𝐵(𝐾) be a multivalued nonexpansive mapping with a fixed point 𝑝. Assume that (i) 0≤𝛼_𝑛, 𝛽_𝑛<1, (ii) 𝛽_𝑛→0, and (iii) ∑𝛼_𝑛𝛽_𝑛=∞. Then, the sequence {𝑥_𝑛} defined by (17) converges strongly to a fixed point of 𝑇.

Panyanak [22] extended the above result of Sastry and Babu [21] to uniformly convex real Banach spaces. He proved the following result.

Theorem P1 (Panyanak [22]). Let 𝐸 be a uniformly convex real Banach space, and let 𝐾 be a nonempty, compact, and convex subset of 𝐸 and 𝑇∶𝐾→𝐶𝐵(𝐾) a multivalued nonexpansive mapping with a fixed point 𝑝. Assume that (i) 0≤𝛼_𝑛, 𝛽_𝑛<1, (ii) 𝛽_𝑛→0, and (iii) ∑𝛼_𝑛𝛽_𝑛=∞. Then, the sequence {𝑥_𝑛} defined by (17) converges strongly to a fixed point of 𝑇.

Panyanak [22] also modified the iteration schemes of Sastry and Babu [21]. Let 𝐾 be a nonempty, closed, and convex subset of a real Banach space, and let 𝑇∶𝐾→𝑃(𝐾) be a multivalued mapping such that 𝐹(𝑇) is a nonempty proximinal subset of 𝐾.

The sequence of Mann iterates is defined by 𝑥₀∈𝐾, 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑦_𝑛,𝛼_𝑛∈[𝑎,𝑏],0<𝑎<𝑏<1, (18) where 𝑦_𝑛∈𝑇𝑥_𝑛 is such that ‖𝑦_𝑛−𝑢_𝑛‖=𝑑(𝑢_𝑛,𝑇𝑥_𝑛) and 𝑢_𝑛∈𝐹(𝑇) is such that ‖𝑥_𝑛−𝑢_𝑛‖=𝑑(𝑥_𝑛,𝐹(𝑇)).

The sequence of Ishikawa iterates is defined by 𝑥₀∈𝐾, 𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑧_𝑛,𝛽_𝑛∈[𝑎,𝑏],0<𝑎<𝑏<1, (19) where 𝑧_𝑛∈𝑇𝑥_𝑛 is such that ‖𝑧_𝑛−𝑢_𝑛‖=𝑑(𝑢_𝑛,𝑇𝑥_𝑛) and 𝑢_𝑛∈𝐹(𝑇) is such that ‖𝑥_𝑛−𝑢_𝑛‖=𝑑(𝑥_𝑛,𝐹(𝑇)). The sequence is defined iteratively by the following way 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑧^′_𝑛,𝛼_𝑛∈[𝑎,𝑏],0<𝑎<𝑏<1, (20) where 𝑧^′_𝑛∈𝑇𝑦_𝑛 is such that ‖𝑧^′_𝑛−𝑣_𝑛‖=𝑑(𝑣_𝑛,𝑇𝑦_𝑛) and 𝑣_𝑛∈𝐹(𝑇) is such that ‖𝑦_𝑛−𝑣_𝑛‖=𝑑(𝑦_𝑛,𝐹(𝑇)). Before we state his theorem, we need the following definition.

Definition 2. A mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is said to satisfy condition (I) if there exists a strictly increasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0, 𝑓(𝑟)>0 for all 𝑟∈(0,∞) such that 𝑑(𝑥,𝑇(𝑥))≥𝑓(𝑑(𝑥,𝐹(𝑇))∀𝑥∈𝐷. (21)

Theorem P2 (Panyanak [22]). Let 𝐸 be a uniformly convex real Banach space, let 𝐾 be a nonempty, closed, bounded, and convex subset of 𝐸, and let 𝑇∶𝐾→𝑃(𝐾) be a multivalued nonexpansive mapping that satisfies condition (I). Assume that (i) 0≤𝛼_𝑛<1 and (ii) ∑𝛼_𝑛=∞. Suppose that 𝐹(𝑇) is a nonempty proximinal subset of 𝐾. Then, the sequence {𝑥_𝑛} defined by (18) converges strongly to a fixed point of 𝑇.

Panyanak [22] then asked the following question.

Question (P). Is Theorem P2 true for the Ishikawa iteration defined by (19) and (20)?

For multivalued mappings, the following lemma is a consequence of the definition of Hausdorff metric, as remarked by Nadler [6].

Lemma 3. Let 𝐴,𝐵∈𝐶𝐵(𝑋) and 𝑎∈𝐴. For every 𝛾>0, there exists 𝑏∈𝐵 such that 𝑑(𝑎,𝑏)≤𝐷(𝐴,𝐵)+𝛾. (22)

Recently, Song and Wang [23] modified the iteration process due to Panyanak [22] and improved the results therein. They gave their iteration scheme as follows.

Let 𝐾 be a nonempty, closed, and convex subset of a real Banach space, and let 𝑇∶𝐾→𝐶𝐵(𝐾) be a multivalued mapping. Let 𝛼_𝑛,𝛽_𝑛∈[0,1] and 𝛾_𝑛∈(0,∞) be such that lim_𝑛→∞𝛾_𝑛=0. Choose 𝑥₀∈𝐾, 𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑧_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑢_𝑛, (23) where 𝑧_𝑛∈𝑇𝑥_𝑛 and 𝑢_𝑛∈𝑇𝑦_𝑛 are such that 𝑧_𝑛−𝑢_𝑛≤𝐷𝑇𝑥_𝑛,𝑇𝑦_𝑛+𝛾_𝑛,𝑧_𝑛+1−𝑢_𝑛≤𝐷𝑇𝑥_𝑛+1,𝑇𝑦_𝑛+𝛾_𝑛. (24) They then proved the following result.

Theorem SW (Song and Wang [23]). Let 𝐾 be a nonempty, compact and convex subset of a uniformly convex real Banach space 𝐸. Let 𝑇∶𝐾→𝐶𝐵(𝐾) be a multivalued nonexpansive mapping with 𝐹(𝑇)≠∅ satisfying 𝑇(𝑝)={𝑝} for all 𝑝∈𝐹(𝑇). Assume that (i) 0≤𝛼_𝑛, 𝛽_𝑛<1, (ii) 𝛽_𝑛→0, and (iii) ∑𝛼_𝑛𝛽_𝑛=∞. Then, the Ishikawa sequence defined by (23) converges strongly to a fixed point of 𝑇.

More recently, Shahzad and Zegeye [29] extended and improved the results of Sastry and Babu [21], Panyanak [22], and Son and Wang [23] to multivalued quasi-nonexpansive mappings. Also, in an attempt to remove the restriction 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇) in Theorem SW, they introduced a new iteration scheme as follows.

Let 𝐾 be a nonempty, closed, and convex subset of a real Banach space, and let 𝑇∶𝐾→𝑃(𝐾) be a multivalued mapping and 𝑃_𝑇𝑥∶={𝑦∈𝑇𝑥∶‖𝑥−𝑦‖=𝑑(𝑥,𝑇𝑥)}. Let 𝛼_𝑛,𝛽_𝑛∈[0,1]. Choose 𝑥₀∈𝐾, and define {𝑥_𝑛} as follows: 𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑧_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑢_𝑛, (25) where 𝑧_𝑛∈𝑃_𝑇𝑥_𝑛 and 𝑢_𝑛∈𝑃_𝑇𝑦_𝑛. They then proved the following result.

Theorem SZ (Shahzad and Zegeye [29]). Let 𝑋 be a uniformly convex real Banach space, let 𝐾 be a nonempty, closed, and convex subset of 𝑋, and let 𝑇∶𝐾→𝑃(𝐾) be a multivalued mapping with 𝐹(𝑇)≠∅ such that 𝑃_𝑇 is nonexpansive. Let {𝑥_𝑛} be the Ishikawa iterates defined by (25). Assume that 𝑇 satisfies condition (I) and 𝛼_𝑛,𝛽_𝑛∈[𝑎,𝑏]⊂(0,1). Then, {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Remark 4. In recursion formula (16), the authors take 𝑦_𝑛∈𝑇(𝑥_𝑛) such that ‖𝑦_𝑛−𝑥^∗‖=𝑑(𝑥^∗,𝑇𝑥_𝑛). The existence of 𝑦_𝑛 satisfying this condition is guaranteed by the assumption that 𝑇𝑥_𝑛 is proximinal. In general such a 𝑦_𝑛 is extremely difficult to pick. If 𝑇𝑥_𝑛 is proximinal, it is not difficult to prove that it is closed. If, in addition, it is a convex subset of a real Hilbert space, then 𝑦_𝑛 is unique and is characterized by 𝑥^∗−𝑦_𝑛,𝑦_𝑛−𝑢_𝑛≥0∀𝑢_𝑛∈𝑇𝑥_𝑛. (26) One can see from this inequality that it is not easy to pick 𝑦_𝑛∈𝑇𝑥_𝑛 satisfying 𝑦_𝑛−𝑥^∗=𝑑𝑥^∗,𝑇𝑥_𝑛 (27) at every step of the iteration process. So, recursion formula (16) is not convenient to use in any possible application. Also, the recursion formula defined in (23) is not convenient to use in any possible application. The sequences {𝑢_𝑛} and {𝑧_𝑛} are not known precisely. Only their existence is guaranteed by Lemma 3. Unlike as in the case of formula (16), characterizations of {𝑢_𝑛} and {𝑧_𝑛} guaranteed by Lemma 3 are not even known. So, recursion formulas (23) are not really useable.

It is our purpose in this paper to first introduce the important class of multivalued strictly pseudocontractive mappings which is more general than the class of multivalued nonexpansive mappings. Then, we prove strong convergence theorems for this class of mappings. The recursion formula used in our more general setting is of the Krasnoselskii type [30] which is known to be superior (see, e.g., Remark 20) to the recursion formula of Mann [31] or Ishikawa [32]. We achieve these results by means of an incisive result similar to the result of Nadler [6] which we prove in Lemma 7.

2. Preliminaries

In the sequel, we will need the following definitions and results.

Definition 5. Let 𝐻 be a real Hilbert space and let 𝑇 be a multivalued mapping. The multivalued mapping (𝐼−𝑇) is said to be strongly demiclosed at 0 (see, e.g., [27]) if for any sequence {𝑥_𝑛}⊆𝐷(𝑇) such that {𝑥_𝑛} converges strongly to 𝑥^∗ and 𝑑(𝑥_𝑛,𝑇𝑥_𝑛) converges strongly to 0, then 𝑑(𝑥^∗,𝑇𝑥^∗)=0.

Definition 6. Let 𝐻 be a real Hilbert space. A multivalued mapping 𝑇∶𝐷(𝑇)⊆𝐻→𝐶𝐵(𝐻) is said to be 𝑘-strictly pseudocontractive if there exist 𝑘∈(0,1) such that for all 𝑥,𝑦∈𝐷(𝑇) one has 𝐷𝑇𝑥,𝑇𝑦²≤𝑥−𝑦²+𝑘(𝑥−𝑢)−𝑦−𝑣²∀𝑢∈𝑇𝑥,𝑣∈𝑇𝑦. (28) If 𝑘=1 in (28), the mapping 𝑇 is said to be pseudocontractive.
We now prove the following lemma which will play a central role in the sequel.

Lemma 7. Let 𝐸 be a reflexive real Banach space and let 𝐴,𝐵∈𝐶𝐵(𝑋). Assume that 𝐵 is weakly closed. Then, for every 𝑎∈𝐴, there exists 𝑏∈𝐵 such that ‖𝑎−𝑏‖≤𝐷(𝐴,𝐵). (29)

Proof. Let 𝑎∈𝐴 and let {𝜆_𝑛} be a sequence of positive real numbers such that 𝜆_𝑛→0 as 𝑛→∞. From Lemma 3, for each 𝑛≥1, there exists 𝑏_𝑛∈𝐵 such that 𝑎−𝑏_𝑛≤𝐷(𝐴,𝐵)+𝜆_𝑛. (30) It then follows that the sequence {𝑏_𝑛} is bounded. Since 𝐸 is reflexive and 𝐵 is weakly closed, there exists a subsequence {𝑏_𝑛𝑘} of {𝑏_𝑛} that converges weakly to some 𝑏∈𝐵. Now, using inequality (30), the fact that {𝑎−𝑏_𝑛𝑘} converges weakly to 𝑎−𝑏 and 𝜆_𝑛𝑘→0, as 𝑘→∞, it follows that ‖𝑎−𝑏‖≤liminf𝑎−𝑏_𝑛𝑘≤𝐷(𝐴,𝐵). (31) This proves the lemma.

Proposition 8. Let 𝐾 be a nonempty subset of a real Hilbert space 𝐻 and let 𝑇∶𝐾→𝐶𝐵(𝐾) be a multivalued 𝑘-strictly pseudocontractive mapping. Assume that for every 𝑥∈𝐾, the set 𝑇𝑥 is weakly closed. Then, 𝑇 is Lipschitzian.

Proof. Let 𝑥,𝑦∈𝐷(𝑇) and 𝑢∈𝑇𝑥. From Lemma 7, there exists 𝑣∈𝑇𝑦 such that ‖𝑢−𝑣‖≤𝐷𝑇𝑥,𝑇𝑦. (32) Using the fact that 𝑇 is 𝑘-strictly pseudocontractive, and inequality (32), we obtain the following estimates: 𝐷𝑇𝑥,𝑇𝑦²≤𝑥−𝑦²+𝑘(𝑥−𝑢)−𝑦−𝑣²≤𝑥−𝑦+√𝑘𝑥−𝑢−𝑦−𝑣², (33) so that 𝐷𝑇𝑥,𝑇𝑦≤𝑥−𝑦+√𝑘+‖‖≤+√𝑥−𝑦𝑢−𝑣𝑥−𝑦𝑘𝑥−𝑦+𝐷𝑇𝑥,𝑇𝑦. (34) Hence, 𝐷≤√𝑇𝑥,𝑇𝑦1+𝑘1−√𝑘𝑥−𝑦. (35) Therefore, 𝑇 is 𝐿-Lipschitzian with 𝐿=∶(1+√𝑘)/(1−√𝑘).

Remark 9. We note that for a single-valued mapping 𝑇, for each 𝑥∈𝐷(𝑇), the set 𝑇𝑥 is always weakly closed.

We now prove the following lemma which will also be crucial in what follows.

Lemma 10. Let 𝐾 be a nonempty and closed subset of a real Hilbert space 𝐻 and let 𝑇∶𝐾→𝑃(𝐾) be a 𝑘-strictly pseudocontractive mapping. Assume that for every 𝑥∈𝐾, the set 𝑇𝑥 is weakly closed. Then, (𝐼−𝑇) is strongly demiclosed at zero.

Proof. Let {𝑥_𝑛}⊆𝐾 be such that 𝑥_𝑛→𝑥 and 𝑑(𝑥_𝑛,𝑇𝑥_𝑛)→0 as 𝑛→∞. Since 𝐾 is closed, we have that 𝑥∈𝐾. Since, for every 𝑛, 𝑇𝑥_𝑛 is proximinal, let 𝑦_𝑛∈𝑇𝑥_n such that ‖𝑥_𝑛−𝑦_𝑛‖=𝑑(𝑥_𝑛,𝑇𝑥_𝑛). Using Lemma 7, for each 𝑛, there exists 𝑧_𝑛∈𝑇𝑥 such that 𝑦_𝑛−𝑧_𝑛≤𝐷𝑇𝑥_𝑛,𝑇𝑥. (36) We then have 𝑥−𝑧_𝑛≤𝑥−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑧_𝑛≤𝑥−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛+𝐷𝑇𝑥_𝑛≤,𝑇𝑥𝑥−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛+1+√𝑘1−√𝑘𝑥_𝑛−𝑥. (37) Observing that 𝑑(𝑥,𝑇𝑥)≤‖𝑥−𝑧_𝑛‖, it then follows that 𝑑(𝑥,𝑇𝑥)≤𝑥−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛+1+√𝑘1−√𝑘𝑥_𝑛−𝑥. (38) Taking the limit as 𝑛→∞, we have that 𝑑(𝑥,𝑇𝑥)=0. Therefore, 𝑥∈𝑇𝑥, completing the proof.

3. Main Results

We prove the following theorem.

Theorem 11. Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Suppose that 𝑇∶𝐾→𝐶𝐵(𝐾) is a multivalued 𝑘-strictly pseudocontractive mapping such that 𝐹(𝑇)≠∅. Assume that 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇). Let {𝑥_𝑛} be a sequence defined by 𝑥₀∈𝐾, 𝑥_𝑛+1=(1−𝜆)𝑥_𝑛+𝜆𝑦_𝑛, (39) where 𝑦_𝑛∈𝑇𝑥_𝑛 and 𝜆∈(0,1−𝑘). Then, lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0.

Proof. Let 𝑝∈𝐹(𝑇). We have the following well-known identity: 𝑡𝑥+(1−𝑡)𝑦²=𝑡‖𝑥‖²+(1−𝑡)𝑦²−𝑡(1−𝑡)𝑥−𝑦², (40) which holds for all 𝑥,𝑦∈𝐻 and for all 𝑡∈[0,1]. Using inequality (28) and the assumption that 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇), we obtain the following estimates: 𝑥_𝑛+1−𝑝²=(1−𝜆)𝑥_𝑛−𝑝+𝜆𝑦_𝑛−𝑝²=(1−𝜆)𝑥_𝑛−𝑝²+𝜆𝑦_𝑛−𝑝²−𝜆(1−𝜆)𝑥_𝑛−𝑦_𝑛²≤(1−𝜆)𝑥_𝑛−𝑝²+𝜆𝐷𝑇𝑥_𝑛,𝑇𝑝²−𝜆(1−𝜆)𝑥_𝑛−𝑦_𝑛²≤(1−𝜆)𝑥_𝑛−𝑝²+𝜆𝑥_𝑛−𝑝²+𝑘𝑥_𝑛−𝑦_𝑛²−𝜆(1−𝜆)𝑥_𝑛−𝑦_𝑛²=𝑥_𝑛−𝑝²+𝜆𝑘𝑥_𝑛−𝑦_𝑛²−𝜆(1−𝜆)𝑥_𝑛−𝑦_𝑛²=𝑥_𝑛−𝑝²−𝜆(1−𝑘−𝜆)𝑥_𝑛−𝑦_𝑛². (41)(42) It then follows that 𝜆(1−𝑘−𝜆)^∞_𝑛=1𝑥_𝑛−𝑦_𝑛²≤𝑥₀−𝑝² (43) which implies that ^∞_𝑛=1𝑥_𝑛−𝑦_𝑛²<∞. (44) Hence, lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0. Since 𝑦_𝑛∈𝑇𝑥_𝑛, we have that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0.

A mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is called hemicompact if, for any sequence {𝑥_𝑛} in 𝐾 such that 𝑑(𝑥_𝑛,𝑇𝑥_𝑛)→0 as 𝑛→∞, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘→𝑝∈𝐾. We note that if 𝐾 is compact, then every multivalued mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is hemicompact.

We now prove the following corollaries of Theorem 11.

Corollary 12. Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐾→𝐶𝐵(𝐾) be a multivalued 𝑘-strictly pseudocontractive mapping with 𝐹(𝑇)≠∅ such that 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇). Suppose that 𝑇 is hemicompact and continuous. Let {𝑥_𝑛} be a sequence defined by 𝑥₀∈𝐾, 𝑥_𝑛+1=(1−𝜆)𝑥_𝑛+𝜆𝑦_𝑛, (45) where 𝑦_𝑛∈𝑇𝑥_𝑛 and 𝜆∈(0,1−𝑘). Then, the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. From Theorem 11, we have that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0. Since 𝑇 is hemicompact, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘→𝑞 as 𝑘→∞ for some 𝑞∈𝐾. Since 𝑇 is continuous, we also have 𝑑(𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘)→𝑑(𝑞,𝑇𝑞) as 𝑘→∞. Therefore, 𝑑(𝑞,𝑇𝑞)=0 and so 𝑞∈𝐹(𝑇). Setting 𝑝=𝑞 in the proof of Theorem 11, it follows from inequality (42) that lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists. So, {𝑥_𝑛} converges strongly to 𝑞. This completes the proof.

Corollary 13. Let 𝐾 be a nonempty, compact, and convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐾→𝐶𝐵(𝐾) be a multivalued 𝑘-strictly pseudocontractive mapping with 𝐹(𝑇)≠∅ such that 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇). Suppose that 𝑇 is continuous. Let {𝑥_𝑛} be a sequence defined by 𝑥₀∈𝐾, 𝑥_𝑛+1=(1−𝜆)𝑥_𝑛+𝜆𝑦_𝑛, (46) where 𝑦_𝑛∈𝑇𝑥_𝑛 and 𝜆∈(0,1−𝑘). Then, the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. Observing that if 𝐾 is compact, every mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is hemicompact, the proof follows from Corollary 12.

Corollary 14. Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐾→𝐶𝐵(𝐾) be a multivalued nonexpansive mapping such that 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇). Suppose that 𝑇 is hemicompact. Let {𝑥_𝑛} be a sequence defined by 𝑥₀∈𝐾, 𝑥_𝑛+1=(1−𝜆)𝑥_𝑛+𝜆𝑦_𝑛, (47) where 𝑦_𝑛∈𝑇𝑥_𝑛 and 𝜆∈(0,1). Then, the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. Since 𝑇 is nonexpansive and hemicompact, then it is strictly pseudocontractive, hemicompact, and continuous. So, the proof follows from Corollary 12.

Remark 15. In Corollary 12, the continuity assumption on 𝑇 can be dispensed with if we assume that for every 𝑥∈𝐾, 𝑇𝑥 is proximinal and weakly closed. In fact, we have the following result.

Corollary 16. Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐾→𝑃(𝐾) be a multivalued 𝑘-strictly pseudocontractive mapping with 𝐹(𝑇)≠∅ such that for every 𝑥∈𝐾, 𝑇𝑥 is weakly closed and 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇). Suppose that 𝑇 is hemicompact. Let {𝑥_𝑛} be a sequence defined by 𝑥₀∈𝐾, 𝑥_𝑛+1=(1−𝜆)𝑥_𝑛+𝜆𝑦_𝑛, (48) where 𝑦_𝑛∈𝑇𝑥_𝑛 and 𝜆∈(0,1−𝑘). Then, the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. Following the same arguments as in the proof of Corollary 12, we have 𝑥_𝑛𝑘→𝑞 and lim_𝑛→∞𝑑(𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘)=0. Furthermore, from Lemma 10, (𝐼−𝑇) is strongly demiclosed at zero. It then follows that 𝑞∈𝑇𝑞. Setting 𝑝=𝑞 and following the same computations as in the proof of Theorem 11, we have from inequality (42) that lim‖𝑥_𝑛−𝑞‖ exists. Since {𝑥_𝑛𝑘} converges strongly to 𝑞, it follows that {𝑥_𝑛} converges strongly to 𝑞∈𝐹(𝑇), completing the proof.

Corollary 17. Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐾→𝑃(𝐾) be a multivalued 𝑘-strictly pseudocontractive mapping with 𝐹(𝑇)≠∅ such that for every 𝑥∈𝐾, 𝑇𝑥 is weakly closed and 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇). Suppose that 𝑇 satisfies condition (I). Let {𝑥_𝑛} be a sequence defined by 𝑥₀∈𝐾, 𝑥_𝑛+1=(1−𝜆)𝑥_𝑛+𝜆𝑦_𝑛, (49) where 𝑦_𝑛∈𝑇𝑥_𝑛 and 𝜆∈(0,1−𝑘). Then, the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. From Theorem 11, we have that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0. Using the fact that 𝑇 satisfies condition (I), it follows that lim_𝑛→∞𝑓(𝑑(𝑥_𝑛,𝐹(𝑇)))=0. Thus there exist a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} and a sequence {𝑝_𝑘}⊂𝐹(𝑇) such that 𝑥_𝑛𝑘−𝑝_𝑘<12^𝑘∀𝑘. (50) By setting 𝑝=𝑝_𝑘 and following the same arguments as in the proof of Theorem 11, we obtain from inequality (42) that 𝑥_𝑛𝑘+1−𝑝_𝑘≤𝑥_𝑛𝑘−𝑝_𝑘<12^𝑘. (51) We now show that {𝑝_𝑘} is a Cauchy sequence in 𝐾. Notice that 𝑝_𝑘+1−𝑝_𝑘≤𝑝_𝑘+1−𝑥_𝑛𝑘+1+𝑥_𝑛𝑘+1−𝑝_𝑘<12^𝑘+1+12^𝑘<12^𝑘−1. (52) This shows that {𝑝_𝑘} is a Cauchy sequence in 𝐾 and thus converges strongly to some 𝑞∈𝐾. Using the fact that 𝑇 is 𝐿-Lipschitzian and 𝑝_𝑘→𝑞, we have 𝑑𝑝_𝑘,𝑇𝑞≤𝐷𝑇𝑝_𝑘𝑝,𝑇𝑞≤𝐿_𝑘−𝑞, (53) so that 𝑑(𝑞,𝑇𝑞)=0 and thus 𝑞∈𝑇𝑞. Therefore, 𝑞∈𝐹(𝑇) and {𝑥_𝑛𝑘} converges strongly to 𝑞. Setting 𝑝=𝑞 in the proof of Theorem 11, it follows from inequality (42) that lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists. So, {𝑥_𝑛} converges strongly to 𝑞. This completes the proof.

Corollary 18. Let 𝐾 be a nonempty compact convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐾→𝑃(𝐾) be a multivalued 𝑘-strictly pseudocontractive mapping with 𝐹(𝑇)≠∅ such that for every 𝑥∈𝐾, 𝑇𝑥 is weakly closed and 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇). Let {𝑥_𝑛} be a sequence defined by 𝑥₀∈𝐾, 𝑥_𝑛+1=(1−𝜆)𝑥_𝑛+𝜆𝑦_𝑛, (54) where 𝑦_𝑛∈𝑇𝑥_𝑛 and 𝜆∈(0,1−𝑘). Then, the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. From Theorem 11, we have that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0. Since {𝑥_𝑛}⊆𝐾 and 𝐾 is compact, {𝑥_𝑛} has a subsequence {𝑥_𝑛𝑘} that converges strongly to some 𝑞∈𝐾. Furthermore, from Lemma 10, (𝐼−𝑇) is strongly demiclosed at zero. It then follows that 𝑞∈𝑇𝑞. Setting 𝑝=𝑞 and following the same arguments as in the proof of Theorem 11, we have from inequality (42) that lim‖𝑥_𝑛−𝑞‖ exists. Since {𝑥_𝑛𝑘} converges strongly to 𝑞, it follows that {𝑥_𝑛} converges strongly to 𝑞∈𝐹(𝑇). This completes the proof.

Corollary 19. Let 𝐾 be a nonempty, compact, and convex subset of a real Hilbert space 𝐸, and let 𝑇∶𝐾→𝑃(𝐾) be a multivalued nonexpansive mapping. Assume that 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇). Let {𝑥_𝑛} be a sequence defined by 𝑥₀∈𝐾, 𝑥_𝑛+1=(1−𝜆)𝑥_𝑛+𝜆𝑦_𝑛, (55) where 𝑦_𝑛∈𝑇𝑥_𝑛 and 𝜆∈(0,1). Then, the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Remark 20. Recursion formula (39) of Theorem 11 is the Krasnoselskii type (see, e.g., [30]) and is known to be superior than the recursion formula of the Mann algorithm (see, e.g., Mann [31]) in the following sense.(i)Recursion formula (39) requires less computation time than the Mann algorithm because the parameter 𝜆 in formula (39) is fixed in (0,1−𝑘), whereas in the algorithm of Mann, 𝜆 is replaced by a sequence {𝑐_𝑛} in (0,1) satisfying the following conditions: ∑^∞_𝑛=1𝑐_𝑛=∞ and lim𝑐_𝑛=0. The 𝑐_𝑛 must be computed at each step of the iteration process.(ii)The Krasnoselskii-type algorithm usually yields rate of convergence as fast as that of a geometric progression, whereas the Mann algorithm usually has order of convergence of the form 𝑜(1/𝑛).

Remark 21. Any consideration of the Ishikawa iterative algorithm (see, e.g., [32]) involving two parameters (two sequences in (0,1)) for the above problem is completely undesirable. Moreover, the rate of convergence of the Ishikawa-type algorithm is generally of the form 𝑜(1/√𝑛) and the algorithm requires a lot more computation than even the Mann process. Consequently, the question asked in [22], Question (P) above, whether an Ishikawa-type algorithm will converge (when it was already known that a Mann-type process converges) has no merit.

Remark 22. Our theorem and corollaries improve convergence theorems for multivalued nonexpansive mappings in [21–23, 25, 26, 28] in the following sense.(i)In our algorithm, 𝑦_𝑛∈T𝑥_𝑛 is arbitrary and does not have to satisfy the very restrictive condition ‖𝑦_𝑛−𝑥^∗‖=𝑑(𝑥^∗,𝑇𝑥_𝑛) in recursion formula (16), and similar restrictions in recursion formula (17). These restrictions on 𝑦_𝑛 depend on 𝑥^∗, a fixed point that is being approximated. (ii)The algorithms used in our theorem and corollaries which are proved for the much larger class of multivalued strict pseudocontractions are of the Krasnoselskii type.

Remark 23. In [29], the authors replace the condition 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇) with the following two restrictions: (i) on the sequence {𝑦_𝑛}:𝑦_𝑛∈𝑃_𝑇𝑥_𝑛, for example, 𝑦_𝑛∈𝑇𝑥_𝑛 and ‖𝑦_𝑛−𝑥_𝑛‖=𝑑(𝑥_𝑛,𝑇𝑥_𝑛). We observe that if 𝑇𝑥_𝑛 is a closed convex subset of a real Hilbert space, then 𝑦_𝑛 is unique and is characterized by 𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑢_𝑛≥0∀𝑢_𝑛∈𝑇𝑥_𝑛; (56) (ii) on 𝑃_𝑇: the authors demand that 𝑃_𝑇 be nonexpansive. So, the first restriction makes the recursion formula difficult to use in any possible application, while the second restriction reduces the class of mappings to which the results are applicable. This is the price to pay for removing the condition 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇).

Remark 24. Corollary 12 is an extension of Theorem 12 of Browder and Petryshyn [19] from single-valued to multivalued strictly pseudocontractive mappings.

Remark 25. A careful examination of our proofs in this paper reveals that all our results have carried over to the class of multivalued quasinonexpansive mappings.

Remark 26. The addition of bounded error terms to the recursion formula (39) leads to no generalization.

We conclude this paper with examples where for each 𝑥∈𝐾, 𝑇𝑥 is proximinal and weakly closed.

Example 27. Let 𝑓∶ℝ→ℝ be an increasing function. Define 𝑇∶ℝ→2^ℝ by 𝑇𝑥=𝑓(𝑥−),𝑓(𝑥+)∀𝑥∈ℝ, (57) where 𝑓(𝑥−)∶=lim_{𝑦→𝑥−}𝑓(𝑦) and 𝑓(𝑥+)∶=lim_𝑦→𝑥+𝑓(𝑦). For every 𝑥∈ℝ, 𝑇𝑥 is either a singleton or a closed and bounded interval. Therefore, 𝑇𝑥 is always weakly closed and convex. Hence, for every 𝑥∈ℝ, the set 𝑇𝑥 is proximinal and weakly closed.

Example 28. Let 𝐻 be a real Hilbert space, and let 𝑓∶𝐻→ℝ be a convex continuous function. Let 𝑇∶𝐻→2^𝐻 be the multivalued mapping defined by 𝑇𝑥=𝜕𝑓(𝑥)∀𝑥∈𝐻, (58) where 𝜕𝑓(𝑥) is the subdifferential of 𝑓 at 𝑥 which is defined by 𝜕𝑓(𝑥)=𝑧∈𝐻∶𝑧,𝑦−𝑥≤𝑓𝑦−𝑓(𝑥)∀𝑦∈𝐻. (59) It is well known that for every 𝑥∈𝐻, 𝜕𝑓(𝑥) is nonempty, weakly closed, and convex. Therefore, since 𝐻 is a real Hilbert space, it then follows that for every 𝑥∈𝐻, the set 𝑇𝑥 is proximinal and weakly closed.

The condition 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇) which is imposed in all our theorems of this paper is not crucial. Our emphasis in this paper is to show that a Krasnoselskii-type sequence converges. It is easy to construct trivial examples for which this condition is satisfied. We do not do this. Instead, we show how this condition can be replaced with another condition which does not assume that the multivalued mapping is single-valued on the nonempty fixed point set. This can be found in the paper by Shahzad and Zegeye [29].

Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space, let 𝑇∶𝐾→𝑃(𝐾) be a multivalued mapping, and let 𝑃_𝑇∶𝐾→𝐶𝐵(𝐾) be defined by 𝑃_𝑇(𝑥)∶=𝑦∈𝑇𝑥∶𝑥−𝑦=𝑑(𝑥,𝑇𝑥). (60) We will need the following result.

Lemma 29 (Song and Cho [33]). Let 𝐾 be a nonempty subset of a real Banach space, and let 𝑇∶𝐾→𝑃(𝐾) be a multivalued mapping. Then, the following are equivalent:(i)𝑥^∗∈𝐹(𝑇); (ii)𝑃_𝑇(𝑥^∗)={𝑥^∗}; (iii)𝑥^∗∈𝐹(𝑃_𝑇).Moreover, 𝐹(𝑇)=𝐹(𝑃_𝑇).

Remark 30. We observe from Lemma 29 that if 𝑇∶𝐾→𝑃(𝐾)  is any multivalued mapping with 𝐹(𝑇)≠∅, then the corresponding multivalued mapping 𝑃_𝑇 satisfies 𝑃_𝑇(𝑝)={𝑝} for all 𝑝∈𝐹(𝑃_𝑇), condition imposed in all our theorems and corollaries. Consequently, examples of multivalued mappings 𝑇∶𝐾→𝐶𝐵(𝐾) satisfying the condition 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇) abound.

Furthermore, we now prove the following theorem where we dispense with the condition 𝑇𝑝={𝑝} for all 𝑝∈𝐹(𝑇).

Theorem 31. Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐾→𝑃(𝐾) be a multivalued mapping such that 𝐹(𝑇)≠∅. Assume that 𝑃_𝑇 is 𝑘-strictly pseudocontractive. Let {𝑥_𝑛} be a sequence defined iteratively from arbitrary point 𝑥₁∈𝐾 by 𝑥_𝑛+1=(1−𝜆)𝑥_𝑛+𝜆𝑦_𝑛, (61) where 𝑦_𝑛∈𝑃_𝑇(𝑥_𝑛) and 𝜆∈(0,1−𝑘). Then, lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0.

Proof. Let 𝑝∈𝐹(𝑇). We have the following well-known identity: 𝑡𝑥+(1−𝑡)𝑦²=𝑡‖𝑥‖²+(1−𝑡)𝑦²−𝑡(1−𝑡)𝑥−𝑦², (62) which holds for all 𝑥,𝑦∈𝐻 and for all 𝑡∈[0,1]. Using recursion formula (61), the identity (62), the fact that 𝑃_𝑇 is 𝑘-strictly pseudocontractive, and Lemma 29, we obtain the following estimates: 𝑥_𝑛+1−𝑝²=(1−𝜆)𝑥_𝑛−𝑝+𝜆𝑦_𝑛−𝑝²=(1−𝜆)𝑥_𝑛−𝑝²+𝜆𝑦_𝑛−𝑝²−𝜆(1−𝜆)𝑥_𝑛−𝑦_𝑛²≤(1−𝜆)𝑥_𝑛−𝑝²+𝜆𝐷𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑝²−𝜆(1−𝜆)𝑥_𝑛−𝑦_𝑛²≤(1−𝜆)𝑥_𝑛−𝑝²+𝜆𝑥_𝑛−𝑝²+𝑘𝑥_𝑛−𝑦_𝑛²−𝜆(1−𝜆)𝑥_𝑛−𝑦_𝑛²=𝑥_𝑛−𝑝²+𝜆𝑘𝑥_𝑛−𝑦_𝑛²−𝜆(1−𝜆)𝑥_𝑛−𝑦_𝑛²=𝑥_𝑛−𝑝²−𝜆(1−𝑘−𝜆)𝑥_𝑛−𝑦_𝑛². (63) It then follows that 𝜆(1−𝑘−𝜆)^∞_𝑛=1𝑥_𝑛−𝑦_𝑛²≤𝑥₀−𝑝², (64) which implies that ^∞_𝑛=1𝑥_𝑛−𝑦_𝑛²<∞. (65) Hence, lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0. Since 𝑦_𝑛∈𝑃_𝑇(𝑥_𝑛) (and hence, 𝑦_𝑛∈𝑇𝑥_𝑛), we have that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0, completing the proof.

We conclude this paper with examples of multivalued mappings 𝑇 for which 𝑃_𝑇 is strictly pseudocontractive, a condition assumed in Theorem 31. Trivially, every nonexpansive mapping is strictly pseudocontractive.

Example 32. Let 𝐻=ℝ, with the usual metric and 𝑇∶ℝ→𝐶𝐵(ℝ) be the multivalued mapping defined by 𝑥𝑇𝑥=0,2(),𝑥,𝑥∈0,∞2,0,𝑥∈(−∞,0]. (66) Then 𝑃_𝑇 is strictly pseudocontractive. In fact, 𝑃_𝑇𝑥={𝑥/2} for all 𝑥∈ℝ.

Example 33. The following example is given in Shahzad and Zegeye [29]. Let 𝐾 be nonempty subset of a normed space 𝐸. A multivalued mapping 𝑇∶𝐾→𝐶𝐵(𝐸) is called ∗-nonexpansive (see, e.g., [34]) if for all 𝑥,𝑦∈𝐾 and 𝑢_𝑥∈𝑇𝑥 with ‖𝑥−𝑢_𝑥‖=𝑑(𝑥,𝑇𝑥), there exists 𝑢_𝑦∈𝑇𝑦 with ‖𝑦−𝑢_𝑦‖=𝑑(𝑦,𝑇𝑦) such that 𝑢_𝑥−𝑢_𝑦≤𝑥−𝑦. (67) It is clear that if 𝑇 is ∗-nonexpansive, then 𝑃_𝑇 is nonexpansive and hence, strictly pseudocontractive. We also note that ∗-nonexpansiveness is different from nonexpansiveness for multivalued mappings. Let K = [0,+∞), and let 𝑇 be defined by 𝑇𝑥=[𝑥,2𝑥] for 𝑥∈𝐾. Then, 𝑃_𝑇(𝑥)={𝑥} for 𝑥∈𝐾 and thus it is nonexpansive and hence strictly pseudocontractive. Note also that 𝑇 is ∗-nonexpansive but is not nonexpansive (see [35]).

Acknowledgment

The authors thank the referees for their comments and remarks that helped to improve the presentation of this paper.